On the stability of higher ring left derivations

In this note, we investigate the Hyers-Ulam, the Isac and Rassias-type stability and the Bourgin-type superstability of a functional inequality corresponding to the following functional equation: hn(xy)=∑i+j=ni≤j[hi(x)hj(y)+cijhi(y)hj(x)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h_n}\left( {xy} \right) = \sum\limits_{\begin{array}{*{20}{c}} {i + j = n} \\ {i \leqslant j} \end{array}} {\left[ {{h_i}\left( x \right){h_j}\left( y \right) + {c_{ij}}{h_i}\left( y \right){h_j}\left( x \right)} \right]} $$\end{document}, where cij={1ifi≠j,0ifi=j.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_{ij}} = \left\{ {\begin{array}{*{20}{c}} 1&{if\;i \ne j,} \\ 0&{if\;i = j.} \end{array}} \right.$$\end{document}